In the semicircle find C and D in terms of A and B Prove tha

In the semicircle find C and D in terms of A and B. Prove that C middot D = 0 (they meet at right angles).

Solution

WE have

B+D=A

So, D=A-B

Now,

C+D=2A

C=2A-D=2A-(A-B)=A+B

Note that, A,B,C,D here are vectors and we are using triangle law of vector addition.

C.D=(A+B).(A-B)=A.A-A.B+B.A-B.B=||A||^2-||B||^2=0

||A||=||B||= radius of given semicircle.